Angle Subtended by a Chord at a Point
Take a line segment PQ and a point R not on the line containing PQ. Join PR and QR (see Fig. 10.10).
Then ∠ PRQ is called the angle subtended by the line segment PQ at the point R. What are angles POQ, PRQ and PSQ called in Fig. 10.11?
∠ POQ is the angle subtended by the chord PQ at the centre O, ∠ PRQ and ∠ PSQ are respectively the angles subtended by PQ at points R and S on the major and minor arcs PQ.
Let us examine the relationship between the size of the chord and the angle subtended by it at the centre.
You may see by drawing different chords of a circle and angles subtended by them at the centre that the longer is the chord, the bigger will be the angle subtended by it at the centre.
What will happen if you take two equal chords of a circle? Will the angles subtended at the centre be the same or not?
Draw two or more equal chords of a circle and measure the angles subtended by them at the centre (see Fig.10.12).
You will find that the angles subtended by them at the centre are equal. Let us give a proof of this fact.
`color{blue}("Theorem 10.1 :")` `"Equal chords of a circle subtend equal angles at the centre."`
`"Proof :"` You are given two equal chords AB and CD of a circle with centre O (see Fig.10.13). You want to prove that ∠ AOB = ∠ COD.
In triangles AOB and COD,
`color{blue}(OA = OC ("Radii of a circle"))`
`color{red}(OB = OD ("Radii of a circle"))`
`color{green}(AB = CD ("Given"))`
`color{blue}("Therefore," Delta AOB ≅ D COD ("SSS rule"))`
This gives `∠ AOB = ∠ COD`
(Corresponding parts of congruent triangles)
`color{green}("Remark :")` For convenience, the abbreviation CPCT will be used in place of ‘Corresponding parts of congruent triangles’, because we use this very frequently as you will see.
Now if two chords of a circle subtend equal angles at the centre, what can you say about the chords? Are they equal or not? Let us examine this by the following activity:
Take a tracing paper and trace a circle on it. Cut it along the circle to get a disc. At its centre O, draw an angle AOB where A, B are points on the circle.
Make another angle POQ at the centre equal to ∠AOB. Cut the disc along AB and PQ (see Fig. 10.14). You will get two segments ACB and PRQ of the circle. If you put one on the other, what do you observe? They cover each other, i.e., they are congruent.
So AB = PQ.
Though you have seen it for this particular case, try it out for other equal angles too. The chords will all turn out to be equal because of the following theorem:
`"Theorem 10.2 :"` `"If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal."`
The above theorem is the converse of the Theorem 10.1. Note that in Fig. 10.13, if you take ∠ AOB = ∠ COD, then
`color{blue}(Delta AOB ≅ D COD )`
So, AB = CD
Then ∠ PRQ is called the angle subtended by the line segment PQ at the point R. What are angles POQ, PRQ and PSQ called in Fig. 10.11?
∠ POQ is the angle subtended by the chord PQ at the centre O, ∠ PRQ and ∠ PSQ are respectively the angles subtended by PQ at points R and S on the major and minor arcs PQ.
Let us examine the relationship between the size of the chord and the angle subtended by it at the centre.
You may see by drawing different chords of a circle and angles subtended by them at the centre that the longer is the chord, the bigger will be the angle subtended by it at the centre.
What will happen if you take two equal chords of a circle? Will the angles subtended at the centre be the same or not?
Draw two or more equal chords of a circle and measure the angles subtended by them at the centre (see Fig.10.12).
You will find that the angles subtended by them at the centre are equal. Let us give a proof of this fact.
`color{blue}("Theorem 10.1 :")` `"Equal chords of a circle subtend equal angles at the centre."`
`"Proof :"` You are given two equal chords AB and CD of a circle with centre O (see Fig.10.13). You want to prove that ∠ AOB = ∠ COD.
In triangles AOB and COD,
`color{blue}(OA = OC ("Radii of a circle"))`
`color{red}(OB = OD ("Radii of a circle"))`
`color{green}(AB = CD ("Given"))`
`color{blue}("Therefore," Delta AOB ≅ D COD ("SSS rule"))`
This gives `∠ AOB = ∠ COD`
(Corresponding parts of congruent triangles)
`color{green}("Remark :")` For convenience, the abbreviation CPCT will be used in place of ‘Corresponding parts of congruent triangles’, because we use this very frequently as you will see.
Now if two chords of a circle subtend equal angles at the centre, what can you say about the chords? Are they equal or not? Let us examine this by the following activity:
Take a tracing paper and trace a circle on it. Cut it along the circle to get a disc. At its centre O, draw an angle AOB where A, B are points on the circle.
Make another angle POQ at the centre equal to ∠AOB. Cut the disc along AB and PQ (see Fig. 10.14). You will get two segments ACB and PRQ of the circle. If you put one on the other, what do you observe? They cover each other, i.e., they are congruent.
So AB = PQ.
Though you have seen it for this particular case, try it out for other equal angles too. The chords will all turn out to be equal because of the following theorem:
`"Theorem 10.2 :"` `"If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal."`
The above theorem is the converse of the Theorem 10.1. Note that in Fig. 10.13, if you take ∠ AOB = ∠ COD, then
`color{blue}(Delta AOB ≅ D COD )`
So, AB = CD